All Questions
927 questions
2
votes
1
answer
232
views
How to discretize a non-linear PDE with boundary conditions and intial value
Consider this non linear PDE:
$$u_t + c(u^2)_x =\alpha u_{xx} \text{ , } -1<x<1 , t>0 $$
with
$$u(-1,t) = g_L(t) \text{ , } u(1,t) = g_R(t) \text{ and } u(x,0)=f(x)
$$
where the 3 functions(...
- 146
2
votes
1
answer
220
views
How to demonstrate the order of convergence of FTBS method for solving a hyperbolic PDE
consider the Purely hyperbolic model problem
$$u_t+au_x=0$$
$$u(-1,t)=u(1,t) \text{ (periodic boundary)}$$
$$u(x,0)=f(x)$$
with $f(y)=\sin(2\pi y)$. Furthermore the exact solution is given by $u(x,t)=...
- 255
2
votes
1
answer
389
views
Solving the 2D Rectangular Waveguide PDE with a Neumann boundary condition for TE modes
I am trying to find the possible modes of a 2d rectangular waveguide by solving the equation,
$$\Big(\frac{\partial^2}{\partial x^2} + \frac{\partial^2}{\partial y^2} + \gamma^2 \Big)\psi = 0$$
where $...
- 123
1
vote
0
answers
109
views
Why is this Advection-convection model with insulating boundary losing mass?
I am trying to model a 1-d advection-convection numerically, using an upwind scheme. I'm using the following equation to calculate the value of internal cells:
$$C_x^{t+1} = C_x^{t} + D\frac{\Delta t}{...
- 119
4
votes
1
answer
139
views
Solving geodesics on triangular meshes gives negative distances
I have implemented the heat method for geodesics:
https://www.cs.cmu.edu/~kmcrane/Projects/HeatMethod/paperCACM.pdf
When I run it I am getting a solution that, visually, seems correct:
In this image, ...
- 379
1
vote
0
answers
150
views
Comparison between stability and accuracy of various Finite Difference schemes
I`m Analyzing the 1D convection equation (PDE) for stability, consistency, and accuracy.
I know Both upwind schemes (explicit and implicit) show better stability with a higher number of waves for ...
5
votes
0
answers
450
views
Stability of Crank-Nicholson for advection diffusion equation for spatial discretization other than finite differences second-order centered
Crank Nicholson is a time discretization method (see 4th equation here). From what I see around, you can use different space discretization, such as Finite elements. But for the linear advection-...
- 445
-3
votes
1
answer
173
views
How to avoid negative concentration from numerical solution using FDM scheme?
$\frac{\partial C}{\partial t} + u \frac{\partial C}{\partial x} + w \frac{\partial C}{\partial x} = D \left(\frac{\partial^2C}{\partial x^2}+\frac{\partial^2C}{\partial y^2}\right)-C \cdot \left(\...
- 1
1
vote
0
answers
60
views
Stability plot of upward difference implicit time
I am analyzing the stability of 1D convection (advection) equation as shown in the picture. When I derive the equations as shown I want to get rid of the complex number.
I`m asking if those stability ...
0
votes
1
answer
225
views
Discretization of a non-linear ODE using FDM isn't grid indepenent
I am trying to solve the ODE :
$\frac{d^2T}{dx^2} = \omega_1 T+\omega_2 T^2$
+
using different numerical methods. I have tried the following discretizations so far and none of them seem to be grid ...
1
vote
0
answers
76
views
discretization of advection diffusion with variable coefficients
I am looking for help to find a somewhat stable FD numerical scheme for the advection diffusion equation posed on a curve $(x(r),y(r))$. The equation becomes
$$u_t=\alpha(r) u_r +\beta(r) u_{rr}+f(r,t)...
- 11
0
votes
1
answer
118
views
High Running Time and Suboptimal Accuracy of 2D Wave Equation Solver with Finite Differences
Im trying to solve the following 2D wave equation: $$u_{tt} = u_{xx} + u_{yy}, \hspace{3mm} u(x,y,0) = \cos(4 \pi x) \sin(4 \pi y), \hspace{3mm} u_t(x,y,0) = 0$$ with the periodic boundary condition ...
- 123
0
votes
1
answer
116
views
2 point BVP solver: how to compute errors
Background
I am working with chapter 2 in LeVeque's book: https://faculty.washington.edu/rjl/fdmbook/
I have build my own solver in Python to solve the 2 point BVP:
$$
\epsilon u''+u(u'-1) =0 , \\
u(0)...
- 255
0
votes
1
answer
1k
views
Trouble Implementing 1d Wave Equation Finite Difference Solver
Im trying to solve the 1d Wave Equation on $x \in \mathbb{R}, t > 0$: $$u_{tt} = c^2u_{xx}, \hspace{5mm} u(x,0) = \cos(4 \pi x), \hspace{5mm} u_t(x,0) = 0$$ with $c = 1$ and a periodic boundary ...
- 123
2
votes
0
answers
167
views
Divergence on wave equation simulation
I'm currenly working on my own PDE solver for non-linear simulations in python. I've done succesfully simulations for KdV and Fisher's equation, but now I'm playing with second order derivatives in ...
0
votes
1
answer
214
views
Incorporating heat flux into Laplace Equation
I need to find the temperature distribution of a square plate using the Laplace equation by using FDM:
$$ \frac{d^2T}{dx^2} + \frac{d^2T}{dy^2} = 0$$
But there is a heat flux entering from the top ...
- 145
1
vote
0
answers
60
views
Results blow up when number of intervals is increases (Yee algorithm FDTD, dielectric sphere)
I have been trying to write a program that analyses EM wave scattering by a dielectric sphere for a project.
The reference is Sadiku's book Numerical Methods in electromagnetics Edition 3.
Now the ...
- 41
2
votes
0
answers
89
views
Linearising Nonlinear Coupled Partial Differential Equations - Alfvénic Diffusion
I am trying to solve the following coupled partial differential equations with a finite difference scheme:
$$\partial_tf+v\partial_zf+\partial_z\frac{1}{W}\partial_zf=0$$
$$\partial_tW+v\partial_zW-\...
- 21
2
votes
0
answers
81
views
How to solve this boundary value problem which has more unknown than equation on MATLAB
I need your helps about solving the problem below with MATLAB. I am trying to solve 2D Stress Wave Propagation problem by using FDTD(Finite difference time domain) method on the cylindrical coord. I ...
- 21
3
votes
1
answer
151
views
Maintain unitary time evolution for a nonlinear ODE
I want to solve a nonlinear ODE of matrix $A(t)$
$$\mathrm{i}\dot A = A(t)M(t),\:\mathrm{with}\: M(t)=A^\dagger(t)H(t)A(t)$$ where $H(t)$ and hence $M(t)$ are Hermitian. Therefore, I presume the time ...
- 303
2
votes
2
answers
468
views
Two-dimensional heat equation with Neumann boundary conditions: any hope to find an analytical solution?
I am looking for references showing how to analytically solve the heat equation with Neumann boundary conditions in two dimensions.
So far, I have found the problem solved analytically in one ...
- 61
0
votes
1
answer
463
views
Verification of order of convergence of Implicit Euler Method to numerically solve Black-Scholes PDE
I'm trying to verify the order of convergence for implicit Euler method to numerically solve Black-Scholes PDE. Theory says that it should be $O(\Delta t + \Delta S^2).$ My code is working absolutely ...
user36184
0
votes
1
answer
458
views
Implementing Dirichlet BC for the Advection-Diffusion equation using a second-order Upwind Scheme finite difference discretization
i am implementing a Matlab code to solve the following equation numerically :
$$
(\frac{\partial c}{\partial t} =-D_{e} \frac{\partial^2 c}{\partial z^2} +U_{z}\frac{\partial c}{\partial z})
$$
with ...
- 1
0
votes
0
answers
893
views
"This DAE appears to be of index greater than 1" daeic12 (line76) error code
Hi I am trying to solve a set of pde converted into ODE and DAE using central finite difference method. I have used the MATLAB 'solve' command to determine the coefficients of fictitious nodes for ...
0
votes
1
answer
2k
views
Compute 1st derivative with backward difference approximation in python
I am trying to write a function to compute 1st derivative with backward difference approximation. $ u'(x_i) = \frac{u(x_i) - u(x_i - \Delta_x)}{\Delta x} \equiv D_- u(x_i).$
And for the first point, ...
- 9
0
votes
1
answer
219
views
Technique to find the CFL condition using the Galerkin method in space and finite-difference in time?
I am using the Galerkin method (Discontinuous to be precise) to discretize in space the scalar linear wave equation and the explicit second order centered finite difference scheme to discretize in ...
- 155
4
votes
1
answer
923
views
Backwards Difference Implicit Method for Nonlinear Parabolic PDE in Python
Original Stack Overflow Question: https://stackoverflow.com/questions/65683788/indexerror-index-31-is-out-of-bounds-for-axis-1-with-size-31?noredirect=1#comment116218335_65683788
PDE: u_t = u_xx + u(...
1
vote
0
answers
198
views
Integrating a wavelike equation with absorbing boundary conditions
I am trying to numerically solve the following equation:
$\frac{\partial^{2} \phi}{\partial t^{2}}-\frac{\partial^{2} \phi}{\partial x^{2}}+V(x) \phi(x, t)=0$ On some domain, with:
$\phi(x, 0) = I(x)$
...
- 111
7
votes
1
answer
190
views
Which finite difference better approximates $uu'$?
I want to approximate $uu'$ with a finite difference. On the one hand, it seems to be
$$(uu')_i=u_i\frac{u_{i+1}-u_{i-1}}{2\Delta t}=\frac{u_iu_{i+1}-u_iu_{i-1}}{2\Delta t}$$
On the other hand,
$$(uu')...
-1
votes
1
answer
244
views
Numerical solution of the advection equation with Crank–Nicolson finite difference method
I need to implement a numerical scheme for the solution of the one-dimensional advection equation
$$\\\frac{\partial u}{\partial t} + C(x, t) \frac{\partial u}{\partial x} = 0 \\\\$$
$$ \\ C(x,t) = \...
- 1
0
votes
0
answers
41
views
Devising Convergent Numerical Scheme for PDE
I'm currently looking at the PDE
\begin{align*}
u_t + \left[x(1-y) - (1-x)\right]u_x - (1-y) u_y + (z-xy) u_z = (z-xy) u_{xy} - (1-x)u& \\
\end{align*}
with
\begin{align*}
u(x,y,z,0) = 1& \\
...
- 141
2
votes
1
answer
106
views
Fix for FD WENO method for multi-component compressible flows
I'm solving two-dimensional four-component compressible Navier-Stokes equations with finite-difference WENO approach. The equations are pretty standard:
$$
\frac{\partial U}{\partial t} +
\frac{\...
- 347
0
votes
0
answers
277
views
How to make a directed graph symmetric?
Say I have a directed graph given as an adjacency matrix $A$ in CSR format represented by the arrays ia (row indexes) and ja (...
- 645
1
vote
2
answers
501
views
How to apply central difference to viscous fluxes in 2D Navier-Stokes equations?
I'm trying to solve 2D unsteady compressible Navier-Stokes equations with finite-difference or finite-volume method. Here is the system, it's pretty standard:
$$
\frac{\partial U}{\partial t} +
\frac{...
- 347
1
vote
1
answer
181
views
Finite Difference for Advection Equation With Source
I'm trying to find a convergent finite difference scheme for the PDE
\begin{equation}
\begin{split}
u_t + (x-1) u_x &= (x-1)u, \hspace{.5cm} x \in [0,1] \\
u(x,0) &= 1 \\
u(1,t) &= 1. \\
\...
- 141
2
votes
0
answers
60
views
In sights into why higher order finite differencing method leads faster to instability
I was playing around with numerically solving the 1D wave equation with density and stiffness varying with position using central differencing methods and noticed that for certain discretization steps ...
- 470
0
votes
0
answers
52
views
Finding a CFD paper with extra degree of freedom variable in mass conservation
I am trying to find a paper that I saw about a year ago. I am not sure of the actual date of the paper.
I believe it was a finite difference CFD paper. The interesting part of the paper was the ...
1
vote
1
answer
66
views
How I can derive the Neuman boundary condition of this system of hyperbolic equations in 1D?
I would like to research the Neuman boundary that can verify the following problem
$\begin{aligned}
&\text { (} P \text { )}\left\{\begin{array}{l}
\frac{\partial U}{\partial t}(x, t)+A \frac{\...
- 145
0
votes
0
answers
222
views
discretizing advection equation with variable wave speed + stability
I currently have a code that solves $u_t+ cu_x=0$ with periodic boundary conditions, and constant $c$ (using an upwind method). I'm wondering how I would alter this code to solve something of the form
...
- 11
2
votes
2
answers
219
views
Heat equation in non-dimensional form behaving differently than in usual format
Starting from
$$
c_p \frac{\partial u }{\partial t} = k \nabla^2 u
$$
in a one dimensional domain [0,1] where $c_p$ and $k$ are modeling two different materials:
$$
k =
\begin{cases}
1 ~\text{if} ~x &...
- 601
1
vote
2
answers
353
views
(Lack of) Availability of Finite-Difference library for simple 2D PDEs
I would like to solve two types of simple 2D problems, namely the stationary heat equation on an L shaped geometry like this:
And also compute the magnetostactic field in an air gap of the following ...
- 570
2
votes
0
answers
79
views
Haw to apply central difference to viscous flux in energy equation?
In many modern papers Navier-Stokes equations are solved with finite-difference or finite-volume methods using WENO reconstruction for non-viscous fluxes and central differences for viscous ones. It ...
- 347
5
votes
0
answers
192
views
2nd-order TVD criteria for flux-limiter
Consider a nonlinear hyperbolic conservation equation:
$$
\partial_{t}u = -\partial_{x}f(u)
$$
The spatial derivative of $f(u)$ may approximated after a spatial discretization by $x_{j}=j\Delta x$
$$
\...
- 95
3
votes
1
answer
620
views
Dirichlet Boundary Condition finite difference method using sparse-matrix $Ax = b$ system
I am trying to solve the boundary value problem for heat equation:
$$
u_{xx} + u_{yy} = f(x,y)
$$
where the solution $u(x,y) \in [0,1] \times [0,1]$ and the Dirichlet boundary condition $u(x,y) = ...
- 131
2
votes
2
answers
180
views
Simplest way to "upgrade" from Euler equations to Navier-Stokes equations in FV or FD framework
I have quite a lot of experience solving unsteady Euler equations, including multi-component ones, with in house-coded finite-difference and finite-volume methods, including MacCormack and MUSCLE ...
- 347
4
votes
1
answer
117
views
Accuracy of Explicit Euler method (finite difference) decreases as Δx decreases, shouldn't it increase?
The price of a commodity can be described by the Schwartz mean reverting SDE
$$dS = \alpha(\mu-\log S)Sdt + \sigma S dW, \qquad \begin{array}.W = \text{ Standard Brownian motion} \\ \alpha = \text{ ...
- 143
1
vote
1
answer
270
views
How is the mixed 2nd partial derivative simplified to a more efficient form?
I'm implementing the Finite Different 2nd and 3rd derivatives in my research and naturally I'm looking for the most efficient approach.
From https://en.wikipedia.org/wiki/Finite_difference#cite_ref-...
- 11
1
vote
1
answer
81
views
If analytically, a function is not differentiable at a point, does it make sense to write a finite difference code for the function at that point?
What would happen if I wrote a finite-difference code evaluated at a point where the function isn't differentiable analytically?
I'm trying to think analytically vs numerically.
Thanks,
- 13
5
votes
1
answer
3k
views
3D laplacian operator
I have been unable to find the equivalent of the 5-point stencil finite differences for the Laplacian operator.
In 2 dimensions for me it is clear that, using the finite difference method: $$
\nabla_{...
- 53
2
votes
1
answer
85
views
Parity for artificial dissipation term in a finite-difference solution
I have a doubt regarding the signal of the dissipative term in a finite difference solution for an equation of the form
$$
\frac{\partial u}{\partial x}+f(x)=0, u(0)=0
$$
In which $f$ is an odd ...
- 91